A small problem

Infinite products are an attractive part of real analysis which has fallen out of many syllabuses. I am concerned here only with infinite products in which the factors are between 0 and 1. The partial products are positive and decreasing, and so tend to a limit; the product is said to converge if the limit is non-zero, or diverge if it is zero. (Take logarithms to see why.)

For example, the infinite product ∏(1−1/n) (over all n≥2) diverges. Yesterday, in the course of a mistaken calculation (I was calculating the wrong thing), I noticed that ∏(1−1/n2) converges to 1/2.

Problem: What is ∏(1−1/nk) for k≥3?

I would be quite happy to be told that this is well known! As I said, I don’t actually need the answer …

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About Peter Cameron

I count all the things that need to be counted.
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2 Responses to A small problem

  1. Ferdinand Ihringer says:

    Equation (20) in

    http://mathworld.wolfram.com/InfiniteProduct.html

    I know nothing about infinite products. I only had a short look at this MathWorld article.

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